Astronomy & Celestial Mechanics
the sky, computed
Concept

Dark matter halo — where it appears

The extended, roughly spherical mass distribution a galaxy sits inside, holding itself up by random motion rather than by rotation. It contains most of the system's mass and a few per cent of its coherent spin.

Named by 21 essays across 4 fields — each of them below, with the objects they name alongside it.

A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

galaxies · Stellar streams A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

galaxies · Galaxy spin Three mass models, one rotation curve. Rotation curves for three decompositions of the same galaxy, from a disc contributing 30 per cent of the outer rotation to one contributing 95. Each is the quadrature sum of a stellar disc, whose shape is fixed by the light distribution and whose amplitude is an unknown mass-to-light ratio, and a dark halo with parameters of its own. All three reproduce the same flat outer rotation, because the halo's amplitude is adjusted to make up whatever the disc does not supply. They differ in the inner few kiloparsecs, by 27 kilometres a second here — which is more than the measurement error and less than the uncertainty in the disc's own contribution, since that depends on a mass-to-light ratio nobody measures directly. The maximum-disc assumption picks the largest disc consistent with the data, and it is a convention rather than a result.

Three mass models that fit the same curve

A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.

galaxies · Dark matter Galaxy formation is inefficient nearly everywhere. Star formation efficiency — the stellar mass a halo has made, divided by the 0.157 of its mass that is baryons — against halo mass, by abundance matching. The n-th most numerous halo is assigned the n-th most numerous galaxy and nothing else is assumed: the halo count is a Sheth–Tormen mass function integrated from a linear power spectrum, the galaxy count is a measured double Schechter, and the matching is a monotone map between two cumulative counts. The curve peaks at 22 per cent, at a halo mass of 10^11.89 M☉, and falls to 0.6 per cent at the bottom of the range and 0.3 per cent at the top. A halo of the Milky Way's mass, 1.3·10¹² M☉, sits near the peak at 21 per cent — and near the peak means near the best any halo has ever managed. The two sides are two different problems and the figure cannot tell them apart: below the peak the shallow potential lets supernovae drive gas out, above it the gas falling in is shock-heated and cannot cool fast enough. What the abundance-matching assumption cannot show is scatter — it assigns one galaxy mass per halo mass by construction, and the real relation has about 0.15 dex of spread that this method is blind to by definition.

Two counts that are not the same shape

Dark matter halos are counted by a calculation that knows nothing about stars. Galaxies are counted by a survey. Laid on the same axes the two curves disagree at both ends and agree nowhere, and the knee is where the two disagreements hand over.

galaxies · Luminosity function The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

spaceflight · Escape A cusp pulled deeper by the disc inside it. The dark-matter density of a halo of 10¹² solar masses and concentration 10 before and after an exponential disc of 5·10¹⁰ solar masses and scale length 3 kpc settles slowly into its centre, computed shell by shell from the adiabatic invariant r·M(r). Far outside the disc the two profiles coincide — 4.8% apart at 83 kpc — because contraction moves mass inward without changing how much there is. Inside, the dark matter is pulled in after the baryons: at half a disc scale length the density is 2.9 times higher, and at 1 kpc the logarithmic slope goes from −1.09 to −1.25. The cusp that dark-matter-only simulations predict is therefore the shallowest the inner halo of a disc galaxy can be before feedback acts, not the profile feedback starts from. The calculation assumes circular orbits and slow settling, and simulations that include gas find the true response weaker than this.

The halo a galaxy drags inward

The cusp that dark-matter simulations predict is a halo with no galaxy in it. When gas cools and settles into a disc at the centre, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull triples the dark matter at the centre and multiplies tenfold the energy needed to dig a core; in a dwarf it barely registers.

galaxies · Halo profiles A cold sphere collapsing onto the virial theorem. The virial ratio 2K/|W| — twice the kinetic energy over the magnitude of the potential energy — counted over the bound shells, of 400 concentric shells released almost at rest (2K/|W| = 0.1) from a sphere whose density falls as one over the radius, against time in units of the sphere's free-fall time. It starts near zero; overshoots to 1.7 as the collapse reaches the centre, the inner shells first; rebounds and oscillates; and settles to 1.03, the virial value, staying within ten per cent of it from about 4.9 free-fall times on. The shells never collide with anything; each feels only the mass inside it. What damps the oscillation is phase mixing — shells of different energy oscillate with different periods and drift out of step — and what redistributes the energy is the potential changing while they move through it. A system of stars reaches equilibrium this way in a few dynamical times, where encounters between the stars would take far longer than the age of the universe.

An equilibrium reached without a single encounter

Every dynamical mass of a galaxy assumes its stars obey the virial theorem, and no two of those stars have ever exchanged energy. A collapse computed shell by shell shows how a system gets there anyway — in a few free-fall times, at half the size it fell from, by a potential that changes while its orbits are still moving through it.

gravitation · Virial theorem The separation of the Milky Way and Andromeda, run forward. The distance between the Milky Way and Andromeda against time from now, in billions of years, integrated from today's separation of 770 kpc and radial approach of 110 km/s, for four values of Andromeda's sideways (tangential) velocity: 0, 17, 57, 100 km/s. The haloes are extended Hernquist spheres of 1.15 and 1.5 trillion solar masses, and dynamical friction drains the orbit at each passage. At 0 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.0 Gyr. At 17 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.8 Gyr. At 57 km/s the first pericentre is 117 kpc at 4.5 Gyr, and the pair merges at 9.2 Gyr. At 100 km/s the first pericentre is 321 kpc at 4.8 Gyr, and the pair has not merged by 14 Gyr. The measured sideways speed runs from about 17 km/s (Hubble) to about 57 km/s (Gaia), and the outcomes across that range differ by billions of years.

A collision decided by a speed across the sky

Andromeda is coming toward the Milky Way at a hundred and ten kilometres a second, and that part is measured to a few per cent. Whether the two galaxies merge in the next ten billion years depends on how fast Andromeda is also moving sideways — a drift of fifteen millionths of an arcsecond a year, measured twice, with answers of seventeen and fifty-seven kilometres a second. One is a near head-on collision in under five billion years, the other a coin toss.

galaxies · Galaxy interactions The observed acceleration against the acceleration the visible matter explains. The acceleration measured from a rotation curve, v²/r, against the acceleration the stars and gas alone would produce at the same radius, both on logarithmic axes in metres per second squared. The solid curve is the fitting function to the measured relation for 153 disc galaxies, g(obs) = g(bar) / (1 − e^(−√(g(bar)/a₀))) with a₀ = 1.2 × 10⁻¹⁰ m s⁻², and the band is its observed scatter of ±0.13 dex, most of which is measurement error. Above a₀ the relation follows the dashed line of equality, where the baryons account for everything; below it, it bends onto the dotted line g(obs) = √(g(bar)·a₀). At ten times a₀ the observed acceleration exceeds the baryonic by a factor of 1.06; at a₀, by 1.58; at a hundredth of a₀, by 10.5. The discrepancy the dark matter is invoked to explain is a function of one variable, and the variable is an acceleration.

One acceleration, and every galaxy's missing mass

Plot the acceleration measured at each radius of a rotation curve against the acceleration its stars and gas would produce there, for every galaxy at once, and the points fall on a single curve. Where the discrepancy begins is set not by a radius or a mass but by an acceleration.

galaxies · Dark matter Haloes at the centre of a void, and in an average place. The number of dark matter haloes per unit volume and per factor e in mass, against halo mass in solar masses per h, in the universe on average (dashed) and inside a sphere of 5 Mpc/h radius at the centre of a void whose density there is 20 per cent of the mean (solid), from the excursion-set theory at redshift zero. The void's sphere holds 9.2e+12 solar masses per h of matter altogether, and no halo can be heavier than that. At the dwarf end the void holds 38 per cent of the average abundance per unit volume — more than its 20 per cent share of the matter. At the mass of the Milky Way's halo, 10¹² M☉/h, it holds 1.62 per cent. An emptiness of galaxies like the Milky Way is guaranteed by the theory; an emptiness of dwarfs is not.

A void is not short of dwarf haloes

The centre of a large void holds a fifth of the average density of matter and almost no galaxies, and it is natural to read the two facts as one. They are not. The theory of how haloes form says a void turns its matter into dwarf-sized haloes at least as readily as anywhere else — it is short of big haloes, not small ones — so a void empty of dwarf galaxies is a statement about which haloes were allowed to light up.

cosmology · Large-scale structure A flattened shell that pulls nothing inside it. The strength of the gravitational field of a thin homoeoid — the shell between two similar concentric ellipsoids, here with axis ratio 0.5 — along its long axis (solid) and its short axis (dashed), in units of GM/a² against distance from the centre in units of the long semi-axis a, from 20,000 equal masses placed on it. The dotted curve is a point mass. Inside the shell the field is zero along both axes to within 0.0001 — the numerical residue of a finite sampling — however flattened the shell is: this is Newton's theorem for a homoeoid, the flattened generalisation of the spherical shell's empty interior. Outside, the shell does not pull like a point: at 1.2 a along the long axis the field is 1.003 against the point's 0.694, and at 0.60 a along the short axis, just outside the shell there, 0.901. The exterior field remembers the shape; the interior does not know there is one.

The shell can be flattened if it thins at the poles

A spherical shell exerts no force anywhere inside it, and a ring exerts a force that points outward. Between them lies a shape Newton found that keeps the sphere's empty interior however far it is flattened: the homoeoid, a shell bounded by two similar ellipsoids, thick at the equator and thin at the poles in exactly the right proportion. It is why the gravity inside any flattened body can be computed at all — and why the planes of stellar streams measure the shapes of dark haloes.

gravitation · Shell theorem Fitted to one curve, three halo histories agree at the Sun and not at the centre. The dark-matter density from 1 to 15 kpc from the Galactic centre, on logarithmic axes in GeV per cubic centimetre, for three halos fitted to the same flat rotation curve with the same disc of 5·10¹⁰ solar masses: one as dark-matter-only simulations produce it, and two contracted by the disc under different assumptions about the dark matter's orbits. At the Sun's 8.2 kpc they give 0.45, 0.41, 0.40 GeV/cm³, a spread of 12 per cent. At 2 kpc they give 3.2, 4.6, 4.6, a spread of 44 per cent. Inside the region the curve constrains, a fit has no freedom to move dark matter about, and the history drops out; inward of it, where the disc and bulge dominate the curve and the dark matter is a small part of the total, the history decides the answer.

A history the rotation curve absorbs

Whether the Milky Way's dark halo was pulled inward by its disc changes the dark matter at the Sun by half again — if the halo is taken from a simulation. Fit it to the Galaxy's own rotation instead, and the history drops out at the Sun's radius and survives only near the centre. The local density that detection experiments depend on is set by the disc's mass, not by the halo's past.

galaxies · Halo profiles A cusp steepened into a spike by the black hole growing in it. The dark-matter density near the Galaxy's central black hole of 4.3·10⁶ solar masses, in solar masses per cubic parsec on logarithmic axes. The dashed line is the halo's NFW cusp, falling inward as one over the radius. If the black hole grew slowly at the centre of that cusp, the dark matter would have followed it inward on adiabatically shrinking orbits, steepening the cusp inside about a fifth of the black hole's sphere of influence — here 0.4 parsecs, with the sphere of influence at 1.8 — into a spike falling as r to the minus 7/3, measured off the drawn curve. Close in, the dark matter's own annihilation would cap the density at the particle mass over the annihilation rate times the Galaxy's age — for a 100 GeV particle with the thermal cross-section, 2.8·10⁸ solar masses per cubic parsec — the flat plateau. Between the spike's edge and its plateau the density rises by 6 orders of magnitude.

A cusp a black hole would sharpen

A black hole that grows slowly at the centre of a dark-matter cusp drags the dark matter in after it, turning a density that rises as one over the radius into a spike rising as the radius to the minus seven-thirds. Whatever the cusp was, the spike comes out nearly the same — and the annihilation signal it would produce is set less by the particle than by whether the spike survived the stars around it.

galaxies · Halo profiles The number of lit dwarfs at a void's centre, against the date it was reionised. The number of haloes heavier than 3.2·10⁹ M☉/h that had collapsed from the matter of a 5 Mpc/h sphere by the redshift at which that sphere was reionised, on a logarithmic axis, against that redshift — for a sphere at the centre of a void with 20 per cent of the mean density today (solid) and an average sphere of the same size (dashed). If only those haloes could make stars, they are the sphere's dwarf galaxies. Reionised at 5.3: 4.3 in the void sphere, 195 in the average one; reionised at 6: 2.0 in the void sphere, 121 in the average one; reionised at 7.7: 0.22 in the void sphere, 29 in the average one. Each unit of delay in the date multiplies the void's count by 3.2 and the average sphere's by 2.1: the void's dwarfs sit further out in the tail of the distribution, and a tail is steep. A count of a few galaxies at a void centre is a statement about when that void's gas was heated, to within a fraction of a unit of redshift — provided the threshold is known.

A void dated by the dwarfs that lit in time

Reionisation did not heat the universe all at once, and the centres of the large voids were among the last places it reached. A dwarf halo there could make stars only if it had grown heavy enough before its own gas was heated — so the number that did is a clock for one void, and a clock that runs faster the emptier the void.

cosmology · Large-scale structure The angular momentum a halo has and the angular momentum a disc needs. The fraction of the mass with specific angular momentum below j, against j in units of the mean, for a halo's material — the universal profile fitted to simulated haloes, with shape parameter μ of 1.1, 1.25, 1.5 — and for an exponential disc with a flat rotation curve (black). Both have the same mean, as a disc that kept its halo's angular momentum would. They do not have the same shape. Below a tenth of the mean a disc holds 2% of its mass; a halo with μ = 1.25 holds 11%, and one with μ = 1.1, 15%. The halo also stops at a largest j, 4.0 times the mean for μ = 1.25, beyond which the disc still has a thin tail. A disc built from a halo's gas without any exchange of angular momentum would be too concentrated at its centre and cut off at its edge.

A halo with too much slow gas for a disc

A disc's size follows from its halo's average spin, and the average is all the simplest account of disc formation uses. The whole distribution does not match. A halo holds five times as much of its material at low angular momentum as an exponential disc can accommodate, so gas that kept its angular momentum exactly would build a dense centre, a thin body and a hard edge — and the bulgeless discs that are common in the nearby universe say that something removed a fifth to a quarter of the slowest gas before it settled.

galaxies · Galaxy spin The flattening a halo has, and the flattening an orbit feels. Three model haloes with flat rotation curves, each built from thin similar-shell layers whose axis ratio changes with radius (dashed), and the axis ratio a nearly circular orbit at each radius responds to (solid) — the a²-weighted mean over the shells inside the orbit, because a shell exerts no force on anything within it. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere. A halo round inside and flattened to 0.8 beyond about 30 kpc is felt as round by every orbit within 20 kpc — 0.988 at 20 — and at 60 kpc as 0.83, where its own shells are 0.80. A halo flattened inside and round outside is felt as flattened well beyond where its shells have become round. An orbit reports a running average of everything inside it, weighted towards its own radius, and nothing at all about what lies outside.

A shell outside the stream turns nothing

A flattened halo turns the plane of every orbit inside it, and a stream of stars several orbits long records how far. But the turning is set only by the halo inside the orbit — a flattened shell enclosing it exerts no force on it at all — and within that, mostly by the shells just inside. So a stream measures a running average of the halo's shape out to its own radius, a halo round inside and flattened outside looks round to every stream in its core, and reading the shape of each layer back out of the streams costs more precision the further out the layer lies.

gravitation · Shell theorem Two places one tidal dwarf could sit. The radial acceleration relation — observed acceleration against the acceleration the baryons alone produce, logarithmic, in m/s² — with the fitted relation for ordinary disc galaxies (solid) and the line of equality (dashed). The marks are a tidal dwarf galaxy of 10⁹ solar masses of gas and stars in an exponential disc of scale length 1.5 kpc, sampled every half kiloparsec to 6 kpc. If it carries no dark matter, as debris pulled from a disc should not, its points lie on the dashed line (open). If the relation is a law that every galaxy obeys, they lie on the curve (filled), and at the outermost radius the acceleration is 5.6 times the baryons' — a rotation speed 2.37 times higher. Every point is below a₀, where the two predictions are furthest apart.

A galaxy born without a halo

When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference.

galaxies · Dark matter A cold slab collapsing into a spiral. 800 self-gravitating sheets in phase space — position across, velocity up — at times 0, 3, 8, 120 in units where a sheet released from the edge reaches the centre in about 1.4. They start as a cold rectangle, uniform in position, with 10 per cent of the kinetic energy equilibrium needs. The slab falls in, overshoots, and the sheets, each orbiting at a period set by its own energy, wind into a spiral whose arms grow thinner and closer until they can no longer be told apart. By the last panel the system is in a steady state: a dense core and a halo of sheets that were flung out in the first collapse. Nothing in the dynamics is an encounter; every change of a sheet's energy comes from the potential changing in time.

A relaxation that stops before it is finished

A cold, collapsing system of stars reaches virial equilibrium in a few crossing times without any star meeting another, because the potential changing in time reshuffles their energies. Lynden-Bell's theory predicted the state it should reach — the most probable one. Computed exactly, in a slab of self-gravitating sheets where every force is known, the collapse reaches equilibrium and not that state: the fluctuations that drive the mixing are what the mixing destroys, and they die before the job is done, leaving a profile and an ordering of energies that remember how the collapse began.

gravitation · Virial theorem A cluster sinking by Chandrasekhar's formula alone. The orbital radius of a 2.0 × 10⁵ solar-mass globular cluster released on a circular orbit at 1 kpc in the two Fornax halo models, decaying under Chandrasekhar's dynamical friction with a velocity dispersion of 11.7 km/s and a Coulomb logarithm of 3. In the cusp it reaches the centre in 6.7 billion years. In the core the formula gives no arrival within 12 billion years, crossing into the core at 2.8. The formula already slows the cluster inside the core, because the circular speed there drops below the stars' velocity dispersion and few stars move slower than the cluster; but it still lets the cluster creep inward without limit. N-body experiments of clusters sinking into cored haloes find something stronger: the decay stops altogether near the core, which the local formula cannot produce at any setting of its logarithm.

A drag that stops where every orbit keeps time

Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.

gravitation · Dynamical friction Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

galaxies · Stellar streams The clusters scale with the halo and the stars do not. The total mass in a galaxy's globular clusters (lower, with its ±0.3 dex scatter shaded) and its stellar mass (upper), against the mass of its dark-matter halo, all logarithmic. The clusters follow a straight line of slope one, 3.9×10⁻⁵ of the halo, across five and a half decades. The stars follow a curve that bends at both ends round a peak near 5.8×10¹¹ M☉, because star formation is suppressed in small halos and in large ones. Stars outweigh clusters by 12 times in a halo of 10¹⁰ M☉, 879 times at 10¹², 31 times at 3.2×10¹⁴.

A halo weighed by counting its oldest clusters

The globular clusters round a galaxy add up to about four parts in a hundred thousand of the mass of its dark-matter halo — in a dwarf, in the Milky Way, in a giant elliptical and in a cluster of galaxies, across five decades of halo mass. The galaxy's own stars follow no such rule. So a count of a few dozen old star clusters weighs a halo that its stars cannot, and the strangest thing about the rule may be that it is less strange than it looks.

galaxies · Dark matter

Named alongside it

The objects these essays reach for when they reach for this one.

Rotation curveDynamical frictionAdiabatic contractionGlobular clusterHalo mass functionNfw profileProper motionShell theoremStellar streamTidal tailVelocity dispersionVirial theorem

All concepts